Particle Motion(AKA Rectilinear Motion)

Vocabulary• Rectilinear Motion

– Position function– Velocity function

• Instantaneous rate of change (position time)– Speed function

• Absolute value of velocity– Acceleration Function

• Instantaneous rate of change (velocity time)• Speeding up/Slowing down

Particle Motion

• Motion on a line

Moving in a positive direction from the origin

Moving in a negative direction from the origin

Position Function• Horizontal axis:

– time• Vertical Axis:

– position on a line

Moving in a positive direction from the origin

time

position

Moving in a negative direction from the origin

Position function: s(t)s = position (sposition duh!)t = times(t)= position changes as time changes

Example

• Use the position and time graph to describe how the puppy was moving

time

position

Velocity• Rate

– position change vs time change

– Velocity can be positive or negative• positive: going in a

positive direction• negative: going in

a negative direction

18

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2

-2

-4

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-10

p

1 2 3 4 5 6 7 8 9 10 11 12t

position

time

A A

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p

-1 1 2 3 4 5 6 7 8 9 10 11t

v(t) x = 3x2+-34x+76

4

time

Animate Points

Velo

city

Pos

ition

Velocity

• Rate at which a coordinate of a particle changes with time

• s(t) = position with respect to time• Instantaneous velocity at time t is:

dtdststv )(')(

time

position

v(t) = positive – increasing slope – moving in a positive direction

v(t) = negative– decreasing slope – moving in a negative direction

Velocity function

• Velocity is the slope of the position function (change in position /change in time)

• velocity =

– This is instantaneous rate of change (position time)

dtdstv )( )(ts

Position Velocity MeaningPositive Slope Positive y’s moving in a positive direction

Negative slope

Negative y’s Moving in a negative direction

Practice• Let s(t)= t3-6t2 be the position function of a

particle moving along an s-axis were s is in meters and t is in seconds. – Graph the position function– On a number line, trace the path that the particle

took. – Where will the velocity be positive? Negative?– Graph the velocity function– Identify on the velocity function when the particle was

heading in a positive direction and when it was heading in a negative direction.

Example: s(t)= t3-6t2 position

time

23 6)( ttts

time

velocity

tttv 123)( 2

tttv 123)( 2

time

speed

Velocity vs Speed

• Speed is change in position with respect to time in any direction

• Velocity is the change in position with respect to time in a particular direction– Thus – Speed cannot be negative – because

going backwards or forwards is just a distance– Thus – Velocity can be negative – because

we care if we go backwards

Speed

• Absolute Value of Velocity

dtdstv

)(

speedousinstantane

example: • if two particles are moving on the same coordinate line • with velocity of v=5 m/s and v=-5 m/s,• then they are going in opposite directions• but both have a speed of |v|=5 m/s

Practice

• Graph the velocity function • What will the speed function look like?• At what time(s) was the particle heading in

a negative direction? Positive direction?

19163)( 2 tttv

Acceleration

• the rate at which the velocity of a particle changes with respect to time.– If s(t) is the position function of a particle

moving on a coordinate line, then the acceleration of the particle at time t is:

dtdvta )(

2

2

)(")(')( dtsdts

dtdstvtaOR

**The second derivative of the position function!!

Example

• Let s(t) = t3 – 6t2 be the position function of a particle moving along an s-axis where s is in meters and t is in seconds. Find the instantaneous acceleration a(t) and shows the graph of acceleration verses time

tttstv 123)(')( 2 126)('')(')( ttstvta

Speeding Up & Slowing Down

• Speeding up velocity and acceleration are the same sign.

• Slowing down when velocity and acceleration are opposite signs.

Example

• When is s(t) speeding up and slowing down?

position

time

23 6)( ttts

time

velocity

tttv 123)( 2

acceleration

Velocity & Acceleration Functions20

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p

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Animate Points

A AB

Slowing down

Velocity +

Acceleration -

Speeding up

Velocity -

Acceleration -

Slowing down

Velocity -

Acceleration +

Speeding up

Velocity +

Acceleration +

Analyzing MotionGraphically Algebraically Meaning

Position

Velocity

Acceleration

Positive “s” values Positive side of the number line

Negative side of the number line

Negative “s” values

s(t)=velocity.

Look for Critical PtsPostive “v” values0 “v” values (CP)

Negative “v” values

Moving in + directionTurning/stopped

Moving in a – direction

v(t)=accelerationLook for Critical Pts

+ a, + v = speeding up- a, - v = speeding up+ a, - v = slowing down- a, + v = slowing down

ExampleSuppose that the position function of a particle moving on a coordinate line is given by s(t) = 2t3-21t2+60t+3 Analyze the motion of the particle for t>0

Graphically Algebraically Meaning

Pos

ition

Velo

city

Acc

eler

atio

n

0360212)( 23 tttts

Never 0 (t>0), always postive

Always on postive side of number line

060426)()( 2 tttvts0)107(6 2 tt

0)5)(2(6 tt

0 2 5

+ - +0 0

0<t<2 going pos directiont=2 turning2<t<5 going neg. directiont=5 turningt>5 going pos. direction

t=0 t=2t=5

04212)()( ttatv4212 t 5.3t

+ - - +0 2 53.5

va - - + +

0<t<2 slowing down2<t<3.5 speeding up3.5<t<5 slowing down

5<t speeding up

Applications: Gravity•

• s = position (height)• s0= initial height• v0= initial velocity• t = time• g= acceleration due to gravity

– g=9.8 m/s2 (meters and seconds)– g=32 ft/s2 (feet and seconds)

200 2

1 gttvss

s0

Example• Nolan Ryan was capable of throwing a baseball at 150ft/s (more

than 102 miles/hour). Could Nolan Ryan have hit the 208 ft ceiling of the Houston Astrodome if he were capable of giving the baseball an upward velocity of 100 ft/s from a height of 7 ft?

2161007 tts tv 32100 the maximum height occurs when velocity = 0

t=100/32=25/8 seconds

s(25/8)=163.25 feet